# PROD, a multiperiod production model

#

# References:

# Robert Fourer, David M. Gay and Brian W. Kernighan, "A Modeling Language

# for Mathematical Programming." Management Science 36 (1990) 519-554.

###  PRODUCTION SETS AND PARAMETERS  ###

set prd 'products';    # Members of the product group

param pt 'production time' {prd} > 0;

                        # Crew-hours to produce 1000 units

param pc 'production cost' {prd} > 0;

                        # Nominal production cost per 1000, used

                        # to compute inventory and shortage costs

###  TIME PERIOD SETS AND PARAMETERS  ###

param first > 0 integer;

                        # Index of first production period to be modeled

param last > first integer;

                        # Index of last production period to be modeled

set time 'planning horizon' := first..last;

###  EMPLOYMENT PARAMETERS  ###

param cs 'crew size' > 0 integer;

                        # Workers per crew

param sl 'shift length' > 0;

                        # Regular-time hours per shift

param rtr 'regular time rate' > 0;

                        # Wage per hour for regular-time labor

param otr 'overtime rate' > rtr;

                        # Wage per hour for overtime labor

param iw 'initial workforce' >= 0 integer;

                        # Crews employed at start of first period

param dpp 'days per period' {time} > 0;

                        # Regular working days in a production period

param ol 'overtime limit' {time} >= 0;

                        # Maximum crew-hours of overtime in a period

param cmin 'crew minimum' {time} >= 0;

                        # Lower limit on average employment in a period

param cmax 'crew maximum' {t in time} >= cmin[t];

                        # Upper limit on average employment in a period

param hc 'hiring cost' {time} >= 0;

                        # Penalty cost of hiring a crew

param lc 'layoff cost' {time} >= 0;

                        # Penalty cost of laying off a crew

###  DEMAND PARAMETERS  ###

param dem 'demand' {prd,first..last+1} >= 0;

                        # Requirements (in 1000s)

                        # to be met from current production and inventory

param pro 'promoted' {prd,first..last+1} logical;

                        # true if product will be the subject

                        # of a special promotion in the period

###  INVENTORY AND SHORTAGE PARAMETERS  ###

param rir 'regular inventory ratio' >= 0;

                        # Proportion of non-promoted demand

                        # that must be in inventory the previous period

param pir 'promotional inventory ratio' >= 0;

                        # Proportion of promoted demand

                        # that must be in inventory the previous period

param life 'inventory lifetime' > 0 integer;

                        # Upper limit on number of periods that

                        # any product may sit in inventory

param cri 'inventory cost ratio' {prd} > 0;

                        # Inventory cost per 1000 units is

                        # cri times nominal production cost

param crs 'shortage cost ratio' {prd} > 0;

                        # Shortage cost per 1000 units is

                        # crs times nominal production cost

param iinv 'initial inventory' {prd} >= 0;

                        # Inventory at start of first period; age unknown

param iil 'initial inventory left' {p in prd, t in time}

              := iinv[p] less sum {v in first..t} dem[p,v];

                        # Initial inventory still available for allocation

                        # at end of period t

param minv 'minimum inventory' {p in prd, t in time}

              := dem[p,t+1] * (if pro[p,t+1] then pir else rir);

                        # Lower limit on inventory at end of period t

###  VARIABLES  ###

var Crews{first-1..last} >= 0;

                        # Average number of crews employed in each period

var Hire{time} >= 0;    # Crews hired from previous to current period

var Layoff{time} >= 0;  # Crews laid off from previous to current period

var Rprd 'regular production' {prd,time} >= 0;

                        # Production using regular-time labor, in 1000s

var Oprd 'overtime production' {prd,time} >= 0;

                        # Production using overtime labor, in 1000s

var Inv 'inventory' {prd,time,1..life} >= 0;

                        # Inv[p,t,a] is the amount of product p that is

                        # a periods old -- produced in period (t+1)-a --

                        # and still in storage at the end of period t

var Short 'shortage' {prd,time} >= 0;

                        # Accumulated unsatisfied demand at the end of period t

###  OBJECTIVE  ###

minimize cost:

    sum {t in time} rtr * sl * dpp[t] * cs * Crews[t] +

    sum {t in time} hc[t] * Hire[t] +

    sum {t in time} lc[t] * Layoff[t] +

    sum {t in time, p in prd} otr * cs * pt[p] * Oprd[p,t] +

    sum {t in time, p in prd, a in 1..life} cri[p] * pc[p] * Inv[p,t,a] +

    sum {t in time, p in prd} crs[p] * pc[p] * Short[p,t];

                        # Full regular wages for all crews employed, plus

                        # penalties for hiring and layoffs, plus

                        # wages for any overtime worked, plus

                        # inventory and shortage costs

                        # (All other production costs are assumed

                        # to depend on initial inventory and on demands,

                        # and so are not included explicitly.)

###  CONSTRAINTS  ###

rlim 'regular-time limit' {t in time}:

    sum {p in prd} pt[p] * Rprd[p,t] <= sl * dpp[t] * Crews[t];

                        # Hours needed to accomplish all regular-time

                        # production in a period must not exceed

                        # hours available on all shifts

olim 'overtime limit' {t in time}:

    sum {p in prd} pt[p] * Oprd[p,t] <= ol[t];

                        # Hours needed to accomplish all overtime

                        # production in a period must not exceed

                        # the specified overtime limit

empl0 'initial crew level':  Crews[first-1] = iw;

                        # Use given initial workforce

empl 'crew levels' {t in time}:  Crews[t] = Crews[t-1] + Hire[t] - Layoff[t];

                        # Workforce changes by hiring or layoffs

emplbnd 'crew limits' {t in time}:  cmin[t] <= Crews[t] <= cmax[t];

                        # Workforce must remain within specified bounds

dreq1 'first demand requirement' {p in prd}:

    Rprd[p,first] + Oprd[p,first] + Short[p,first]

                             - Inv[p,first,1] = dem[p,first] less iinv[p];

dreq 'demand requirements' {p in prd, t in first+1..last}:

    Rprd[p,t] + Oprd[p,t] + Short[p,t] - Short[p,t-1]

                          + sum {a in 1..life} (Inv[p,t-1,a] - Inv[p,t,a])

                                                  = dem[p,t] less iil[p,t-1];

                        # Production plus increase in shortage plus

                        # decrease in inventory must equal demand

ireq 'inventory requirements' {p in prd, t in time}:

    sum {a in 1..life} Inv[p,t,a] + iil[p,t] >= minv[p,t];

                        # Inventory in storage at end of period t

                        # must meet specified minimum

izero 'impossible inventories' {p in prd, v in 1..life-1, a in v+1..life}:

    Inv[p,first+v-1,a] = 0;

                        # In the vth period (starting from first)

                        # no inventory may be more than v periods old

                        # (initial inventories are handled separately)

ilim1 'new-inventory limits' {p in prd, t in time}:

    Inv[p,t,1] <= Rprd[p,t] + Oprd[p,t];

                        # New inventory cannot exceed

                        # production in the most recent period

ilim 'inventory limits' {p in prd, t in first+1..last, a in 2..life}:

    Inv[p,t,a] <= Inv[p,t-1,a-1];

                        # Inventory left from period (t+1)-p

                        # can only decrease as time goes on

###  DATA  ###

data;

set prd := 18REG 24REG 24PRO ;

param first :=  1 ;

param last  := 13 ;

param life  :=  2 ;

param cs := 18 ;

param sl :=  8 ;

param iw :=  8 ;

param rtr := 16.00 ;

param otr := 43.85 ;

param rir :=  0.75 ;

param pir :=  0.80 ;

param :         pt       pc        cri       crs      iinv   :=

  18REG      1.194     2304.     0.015     1.100      82.0

  24REG      1.509     2920.     0.015     1.100     792.2

  24PRO      1.509     2910.     0.015     1.100       0.0   ;

param :     dpp        ol      cmin      cmax        hc        lc   :=

  1        19.5      96.0       0.0       8.0      7500      7500

  2        19.0      96.0       0.0       8.0      7500      7500

  3        20.0      96.0       0.0       8.0      7500      7500

  4        19.0      96.0       0.0       8.0      7500      7500

  5        19.5      96.0       0.0       8.0     15000     15000

  6        19.0      96.0       0.0       8.0     15000     15000

  7        19.0      96.0       0.0       8.0     15000     15000

  8        20.0      96.0       0.0       8.0     15000     15000

  9        19.0      96.0       0.0       8.0     15000     15000

 10        20.0      96.0       0.0       8.0     15000     15000

 11        20.0      96.0       0.0       8.0      7500      7500

 12        18.0      96.0       0.0       8.0      7500      7500

 13        18.0      96.0       0.0       8.0      7500      7500   ;

param dem (tr) :

          18REG     24REG     24PRO   :=

  1        63.8    1212.0       0.0

  2        76.0     306.2       0.0

  3        88.4     319.0       0.0

  4       913.8     208.4       0.0

  5       115.0     298.0       0.0

  6       133.8     328.2       0.0

  7        79.6     959.6       0.0

  8       111.0     257.6       0.0

  9       121.6     335.6       0.0

 10       470.0     118.0    1102.0

 11        78.4     284.8       0.0

 12        99.4     970.0       0.0

 13       140.4     343.8       0.0

 14        63.8    1212.0       0.0   ;

param pro (tr) :

          18REG     24REG     24PRO   :=

  1           0         1         0

  2           0         0         0

  3           0         0         0

  4           1         0         0

  5           0         0         0

  6           0         0         0

  7           0         1         0

  8           0         0         0

  9           0         0         0

 10           1         0         1

 11           0         0         0

 12           0         0         0

 13           0         1         0

 14           0         1         0   ;

end;